Note that we restrict ourselves to finite surjective morphisms, as opposed to Corvaja-Zannier who consider dominant rational maps of finite degree in their definition of the weak-Hilbert property (see [1]}). By a standard argument using Stein factorization, one can easily show that these two definitions are equivalent (see, for example, the proof of Theorem REF ). In addition, we stress that the weak-Hilbert property is a birational invariant amongst smooth proper varieties (see Proposition REF ). As a consequence, all our results concerning this property automatically propagate between smooth proper varieties within the same birational equivalence class.
